=g1g2324 T1+I12 m23 ds [s-(m1+m2)2][s-(m1-m2)2]s  K1(s/T),

where the factor \(1/(1+I_{12})\)  is introduced to avoid double counting of indistinguishable particles, we have \(I_{12}=1\)  for identical particles, otherwise \(I_{12}=0\) .
Many studies have shown that the hadronic reaction matrix element \(|\mathcal {M}_{12\rightarrow 3}|^2\)  for hadronic reactions without resonance phenomenon is insensitive to changes of \(\sqrt{s}\)  in the considered energy range [1]}, [2]}. In this case, we assume that the transition amplitude \(|\mathcal {M}_{12\rightarrow 3}|^2\)  is nearly constant in the energy range we consider, and the cross section \(\sigma \)  can be written as

=22[s-(m1+m2)2][s-(m1-m2)2] (1g1g2spin|M123|2)IMS1(out)

where the one-particle invariant phase space is given by

IMS1(out)= d4p3 0(p23-m23) 4(p1+p2-p3)=0(s-m23).

In this case, the thermal reaction rate per time and volume can be written as

R123=1163 T1+I spin|M123|2m23 ds[s-(m1+m2)2][s-(m1-m2)2]2s0(s-m23) K1(s/T)
